Financial models with long-tailed distributions and volatility clustering

Classical financial models which assume homoskedasticity and normality cannot explain stylized phenomena such as skewness, heavy tails, and volatility clustering of the empirical asset returns in finance. In 1963, Benoit Mandelbrot first used the stable (or $\alpha$-stable) distribution to model the empirical distributions which have the skewness and heavy-tail property. Since $\alpha$-stable distributions have infinite $p$-th moments for all $p>\alpha$, the tempered stable processes have been proposed for overcoming this limitation of the stable distribution.

On the other hand, GARCH models have been developed to explain the volatility clustering. In the GARCH model, the innovation (or residual) distributions are assumed to be a standard normal distribution, despite the fact that this assumption is often rejected empirically. For this reason, GARCH models with non-normal innovation distribution have been developed.

Many financial models with the stable and the tempered stable distributions together with volatility clustering have been developed and applied to risk management, option pricing, and portfolio selection.

Infinitely divisible distributions

A random variable $Y$ is called infinitely divisible if, for each $n=1,2,\cdots$, there are i.i.d random variables $Y_{n,1},Y_{n,2},\cdots ,Y_{n,n}$ such that

$Y{\stackrel {\mathrm {d} }{=}}\sum _{k=1}^{n}Y_{n,k}$,

where ${\stackrel {\mathrm {d} }{=}}$ denotes equality in distribution.

A Borel measure $\nu$ on $\mathbb {R}$ is called a Levy measure if $\nu ({0})=0$ and

$\int _{\mathbb {R} }(1\wedge |x^{2}|)\nu (dx)<\infty$.

If $Y$ is infinitely divisible, then the characteristic function $\phi _{Y}(u)=E[e^{iuY}]$ is given by

$\phi _{Y}(u)=\exp \left(i\gamma u-{\frac {1}{2}}\sigma ^{2}u+\int _{-\infty }^{\infty }(e^{iux}-1-iux1_{|x|\leq 1})\nu (dx)\right),\sigma \geq 0,~~\gamma \in \mathbb {R}$

where $\sigma \geq 0$, $\gamma \in \mathbb {R}$ and $\nu$ is a Levy measure. Here the triple $(\sigma ^{2},\nu ,\gamma )$ is called a Levy triplet of $Y$. This triplet is unique. Conversely, for any choice $(\sigma ^{2},\nu ,\gamma )$ satisfying the conditions above, there exists an infinitely divisible random variable $Y$ whose characteristic function is given as $\phi _{Y}$.

$\alpha$-Stable distributions

An real-valued random variable $X$ is said to have an $\alpha$-stable distribution if for any $n\geq 2$, there are a positive number $C_{n}$ and a real number $D_{n}$ such that

$X_{1}+\cdots +X_{n}{\stackrel {\mathrm {d} }{=}}C_{n}X+D_{n},$

where $X_{1},X_{2},\cdots ,X_{n}$ are independent and have the same distribution as that of $X$. All stable random variables are infinitely divisible. It is known that $C_{n}=n^{1/\alpha }$ for some $0<\alpha \leq 2$. A stable random variable $X$ with index $\alpha$ is called an $\alpha$-stable random variable.

Let $X$ be an $\alpha$-stable random variable. Then the characteristic function $\phi _{X}$ of $X$ is given by
$\phi _{X}(u)=\left\{{\begin{array}{ll}\exp \left(i\mu u-\sigma ^{\alpha }|u|^{\alpha }\left(1-i\beta {\text{sgn}}(u)\tan \left({\frac {\pi \alpha }{2}}\right)\right)\right)&{\text{if}}~~\alpha \in (0,1)\cup (1,2)\\\exp \left(i\mu u-\sigma |u|\left(1+i\beta {\text{sgn}}(u)\left({\frac {2}{\pi }}\right)\ln(|u|)\right)\right)&{\text{if}}~~\alpha =1\\\exp \left(i\mu u-{\frac {1}{2}}\sigma ^{2}u^{2}\right)&{\text{if}}~~\alpha =2\end{array}}\right.$
for some $\mu \in \mathbb {R}$, $\sigma >0$ and $\beta \in [-1,1]$.

References
[1] B. B. Mandelbrot, New Methods in Statistical Economics, Journal of Political Economy, 71, 421-440, 1963.
[2] Svetlozar. T. Rachev, C. Menn, Frank J. Fabozzi, Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing,John Wiley & Sons, 2005.
[3] Svetlozar. T. Rachev, S. Mitnik, Stable Paretian Models in Finance, John Wiley & Sons, 2000.
[4] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes, Chapman & Hall/CRC.

Tempered stable distributions

An infinitely divisible distribution is called a classical tempered stable (CTS) distribution with parameter $(C_{1},C_{2},\lambda _{+},\lambda _{-},\alpha )$, if its Levy triplet $(\sigma ^{2},\nu ,\gamma )$ is given by $\sigma =0$, $\gamma \in \mathbb {R}$ and $\nu (dx)=\left({\frac {C_{1}e^{-\lambda _{+}x}}{x^{1+\alpha }}}1_{x>0}+{\frac {C_{2}e^{-\lambda _{-}|x|}}{|x|^{1+\alpha }}}1_{x<0}\right)dx,$ where $C_{1},C_{2},\lambda _{+},\lambda _{-}>0$ and $\alpha <2$.

This distribution was first introduced by Koponen [5] under the name of Truncated Levy Flights and has been called the tempered stable or the KoBoL distribution [1]. In particular, if $C_{1}=C_{2}=C>0$, then this distribution is called the CGMY distribution which has been used in Carr et al. [2] for financial modeling.

The characteristic function $\phi _{CTS}$ for a tempered stable distribution is given by

$\phi _{CTS}(u)=\exp \left(iu\mu +C_{1}\Gamma (-\alpha )((\lambda _{+}-iu)^{\alpha }-\lambda _{+}^{\alpha })+C_{2}\Gamma (-\alpha )((\lambda _{-}+iu)^{\alpha }-\lambda _{-}^{\alpha })\right),$

for some $\mu \in \mathbb {R}$. Moreover, $\phi _{CTS}$ can be extended to the region $\{z\in \mathbb {C} :Im(z)\in (-\lambda _{-},\lambda _{+})\}$.

Rosiński [6] generalized the CTS distribution under the name of the tempered stable distribution. The KR distribution, which is a subclass of the Rosiński's generalized tempered stable distributions, is used in finance [4].

An infinitely divisible distribution is called a modified tempered stable (MTS) distribution with parameter $(C,\lambda _{+},\lambda _{-},\alpha )$, if its Levy triplet $(\sigma ^{2},\nu ,\gamma )$ is given by $\sigma =0$, $\gamma \in \mathbb {R}$ and $\nu (dx)=C({\frac {q_{\alpha }(\lambda _{+}|x|)}{x^{\alpha +1}}}1_{x>0}+{\frac {q_{\alpha }(\lambda _{-}|x|)}{|x|^{\alpha +1}}}1_{x<0})dx,$ where $C,\lambda _{+},\lambda _{-}>0,\alpha <2$ and $q_{\alpha }(x)=x^{\frac {\alpha +1}{2}}K_{\frac {\alpha +1}{2}}(x)$. $K_{p}(x)$ is the modified Bessel function of the second kind. The MTS distribution is not included in the class of Rosiński's generalized tempered stable distributions [3].

References
[1] S. I. Boyarchenko, S. Z. Levendorskiǐ, Option pricing for truncated Levy processes, International Journal of Theoretical and Applied Finance, 3 (2000), 3, pp. 549-552.
[2] P. Carr, H. Geman, D. Madan, M. Yor, The Fine Structure of Asset Returns: An Empirical Investigation, Journal of Business, 75 (2002), 2, pp. 305-332.
[3] Y.S. Kim, Svetlozar. T. Rachev, D. M. Chung, The modified tempered stable distribution, GARCH models and option pricing, Probability and Mathematical Statistics, to appear
[4] Y.S. Kim, Svetlozar. T. Rachev, M. L. Bianchi, Frank J. Fabozzi, A New Tempered Stable Distribution and Its Application to Finance, Georg Bol, Svetlozar T. Rachev, and Reinold Wuerth (Eds.), Risk Assessment: Decisions in Banking and Finance, Physika Verlag, Springer 2007.
[5] I. Koponen, Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process, Physical Review E, 52 (1995), pp. 1197-1199.
[6] J. Rosiński, Tempering Stable Processes, Stochastic Processes and their Applications, 117 (2007), 6, pp. 677-707.

Volatility clustering with stable and tempered stable innovation

In order to describe the volatility clustering effect of the return process of an asset, the GARCH model can be used. In the GARCH model, innovation ($~\epsilon _{t}~$) is assumed that $~\epsilon _{t}=\sigma _{t}z_{t}~$, where $z_{t}\sim iid~N(0,1)$ and where the series $\sigma _{t}^{2}$ are modeled by

$\sigma _{t}^{2}=\alpha _{0}+\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}=\alpha _{0}+\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}$

and where $~\alpha _{0}>0~$ and $\alpha _{i}\geq 0,~i>0$.

However, the assumption of $z_{t}\sim iid~N(0,1)$ is often rejected empirically. For that reason, new GARCH models with stable or tempered stable distributed innovation have been developed. GARCH models with $\alpha$-stable innovations are introduced by Menn and Rachev [3,4,5]. Then the GARCH Models with tempered stable innovations have been developed in [1] and [2].

References

[1] Y.S. Kim, Svetlozar. T. Rachev, D.M. Chung, Michele L. Bianchi, The modified tempered stable distribution, GARCH models and option pricing, Probability and Mathematical Statistics, to appear.
[2] Y.S. Kim, Svetlozar. T. Rachev, Michele L. Bianchi, Frank J. Fabozzi, Financial market models with Levy processes and time-varying volatility Journal of Banking & Finance, 32, 7, 1363-1378, 2008.
[3] C. Menn, Svetlozar. T. Rachev, A GARCH Option Pricing Model with $\alpha$-stable Innovations, European Journal of Operational Research, 163, 201-209, 2005.
[4] C. Menn, Svetlozar. T. Rachev, Smoothly Truncated Stable Distributions, GARCH-Models, and Option Pricing, Technical report: Chair of Econometrics, Statistics and Math- ematical Finance School of Economics and Business Engineering University of Karlsruhe, 2005.
[5] Svetlozar. T. Rachev, C. Menn, Frank J. Fabozzi, Fat-Tailed and Skewed Asset Return Distributions: Implications for Risk Management, Portfolio selection, and Option Pricing, John Wiley & Sons, 2005.

See Also

G. Samorodnitsky
Svetlozar. T. Rachev
Aleksander Weron
Finanalytica